EP1400905A1 - Procédé et dispositif pour la détermination adaptive de facteurs de pondération dans le contexte d'une function objective - Google Patents

Procédé et dispositif pour la détermination adaptive de facteurs de pondération dans le contexte d'une function objective Download PDF

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Publication number
EP1400905A1
EP1400905A1 EP03255772A EP03255772A EP1400905A1 EP 1400905 A1 EP1400905 A1 EP 1400905A1 EP 03255772 A EP03255772 A EP 03255772A EP 03255772 A EP03255772 A EP 03255772A EP 1400905 A1 EP1400905 A1 EP 1400905A1
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Prior art keywords
penalty
component
objective function
weight
credit
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German (de)
English (en)
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David Joseph Kropaczek
William Earl Russell Ii
William Charles Cline
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Global Nuclear Fuel Americas LLC
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Global Nuclear Fuel Americas LLC
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G—PHYSICS
    • G21—NUCLEAR PHYSICS; NUCLEAR ENGINEERING
    • G21D—NUCLEAR POWER PLANT
    • G21D3/00—Control of nuclear power plant
    • G21D3/001—Computer implemented control
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06Q—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES; SYSTEMS OR METHODS SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES, NOT OTHERWISE PROVIDED FOR
    • G06Q10/00—Administration; Management
    • G06Q10/04—Forecasting or optimisation specially adapted for administrative or management purposes, e.g. linear programming or "cutting stock problem"
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2111/00—Details relating to CAD techniques
    • G06F2111/04—Constraint-based CAD
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2111/00—Details relating to CAD techniques
    • G06F2111/06—Multi-objective optimisation, e.g. Pareto optimisation using simulated annealing [SA], ant colony algorithms or genetic algorithms [GA]
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02E—REDUCTION OF GREENHOUSE GAS [GHG] EMISSIONS, RELATED TO ENERGY GENERATION, TRANSMISSION OR DISTRIBUTION
    • Y02E30/00—Energy generation of nuclear origin
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02E—REDUCTION OF GREENHOUSE GAS [GHG] EMISSIONS, RELATED TO ENERGY GENERATION, TRANSMISSION OR DISTRIBUTION
    • Y02E30/00—Energy generation of nuclear origin
    • Y02E30/30—Nuclear fission reactors

Definitions

  • a Free Optimization Problem is one for which no constraints exist.
  • a Constraint Optimization Problem includes both constraints and a minimize (or maximize) condition(s) requirement.
  • a Constraint Satisfaction Problem contains only constraints. Solving a CSP means finding one feasible solution within the search space that satisfies the constraint conditions. Solving a COP means finding a solution that is both feasible and optimal in the sense that a minimum (or maximum) value for the desired condition(s) is realized.
  • the solution to such a problem typically involves a mathematical search algorithm, whereby successively improved solutions are obtained over the course of a number of algorithm iterations. Each iteration, which can be thought of as a proposed solution, results in improvement of an objective function.
  • An objective function is a mathematical expression having parameter values of a proposed solution as inputs. The objective function produces a figure of merit for the proposed solution. Comparison of objective function values provides a measure as to the relative strength of one solution versus another.
  • the penalty approach broadens the search space by allowing examination of both feasible and infeasible solutions in an unbiased manner. Broadening the search space during an optimization search often allows local minima to be circumnavigated more readily, thus making for a more effective optimization algorithm.
  • alternate methods for handling constraints such as infeasible solution 'repair' and 'behavioral memory', are based on maintaining or forcing feasibility among solutions that are examined during the optimization search.
  • a mathematical expression is defined for each constraint that quantifies the magnitude of the constraint violation. For the given constraint, a weighting factor then multiplies the result to create an objective function penalty component. Summing all penalty components yields the total penalty. The larger the weighting factor for a given constraint, the greater the emphasis the optimization search will place on resolving violations in the constraint during the optimization search.
  • the simplest penalty function form is the 'death penalty', which sets the value of the weighting factor for each constraint to infinity. With a death penalty the search algorithm will immediately reject any violation of a constraint, which is equivalent to rejecting all infeasible solutions.
  • Static penalties apply a finite penalty value to each constraint defined.
  • a static weighting factor maintains its initial input value throughout the optimization search.
  • Dynamic penalties adjust the initial input value during the course of the optimization search according to a mathematical expression that determines the amount and frequency of the weight change.
  • the form of the penalty functions in a dynamic penalty scheme contains, in addition to the initial static penalty weighting factors (required to start the search), additional parameters that must be input as part of the optimization algorithm.
  • adaptive penalties adjust weight values over the course of an optimization search.
  • the amount and frequency of the weight change is determined by the progress of the optimization search in finding improved solutions.
  • Bean and Hadj-Alouane created the method of Adaptive Penalties (AP), which was implemented in the context of a Genetic Algorithm.
  • AP Adaptive Penalties
  • the population of solutions obtained over a preset number of iterations of the optimization search is examined and the weights adjusted depending on whether the population contains only feasible, infeasible, or a mixture of feasible and infeasible solutions.
  • Coit, Smith, and Tate proposed an adaptive penalty method based on estimating a 'Near Feasibility Threshold' (NFT) for each given constraint.
  • NFT 'Near Feasibility Threshold'
  • the NFT defines a region of infeasible search space just outside of feasibility that the optimization search would then be permitted to explore.
  • Eiben and Hemert developed the Stepwise Adaption of Weights (SAW) method for adapting penalties. In their method, a weighting factor adjustment is made periodically to each constraint that violates in the best solution, thus potentially biasing future solutions away from constraint violations.
  • SAW Stepwise Adaption of Weights
  • Penalty adaption improves over the static and dynamic penalty approaches by attempting to utilize information about the specific problem being solved as the optimization search progresses. In effect, the problem is periodically redefined.
  • a deficiency with the adaptive penalty approach is that the objective function loses all meaning in an absolute sense during the course of an optimization search. In other words, there is no 'memory' that ties the objective function back to the original starting point of the optimization search as exists in a static penalty or dynamic penalty approach.
  • the invention is a method and apparatus for adaptively determining weighting factors within the context of an objective function for handling optimality conditions and constraints within an optimization search.
  • the invention is not dependent on any particular optimization search technique so long as it conforms to a process of iterative improvement and a particular form for the objective function, defined as a sum of credit and penalty components.
  • the credit components represent the optimality conditions for the problem.
  • the penalty components represent the constraint violations for the problem. Initially, each component is made up of a weight multiplied by a mathematical expression, called a term, that quantifies either an optimality condition or a constraint violation.
  • the invention performs an adaptive determination of the set of credit and penalty weights based on the progress of the optimization search.
  • the invention utilizes both static and dynamic representations of the objective function to perform the adaption.
  • the static representation is based on a set of user defined input weighting factors that remain fixed throughout the optimization while the dynamic representation is the 'true' objective function as utilized by the optimization search in assessing solution fitness.
  • the adjustments to the weighting factors are performed during the course of the optimization search.
  • the magnitude of the penalty weight for the 'worst' penalty component is increased while simultaneously decreasing the weight for the remaining penalty and credit components.
  • the worst penalty component is calculated from the product of the penalty weight and the penalty term, where the penalty weights are the initial static values, for example, as input by the user; thus, providing memory to tie the objective function back to the original starting point.
  • the magnitudes of the credit weights for the credit components are increased while maintaining the existing dynamic weight values for the penalty weights.
  • the present invention applies, but is not limited to, a generic definition of an objective function, which is applicable across a wide variety of constraint and optimization problems.
  • the generic objective function is applicable to any large scale, combinatorial optimization problem in discrete or continuous space such as boiler water reactor core design, pressurized water reactor core design, transportation scheduling, resource allocation, etc.
  • the generic objective function is defined as a sum of credit and penalty components.
  • a penalty component includes a penalty term multiplied by an associated penalty weight.
  • a credit component includes a credit term multiplied by an associated credit weight.
  • the credit terms represent the optimality conditions for the problem.
  • the penalty terms represent the constraints for the problem.
  • Each credit term is a mathematical expression that quantifies an optimality condition.
  • Each penalty term is a mathematical expression that quantifies a constraint.
  • Credit and penalty terms may be defined by maximum (i.e. upper bounded) or minimum (i.e. lower bounded) values and can represent scalar or multidimensional values. The only requirements are: 1) the penalty terms must be positive for constraint violations and zero otherwise, and 2) in the absence of constraint violations, the credit terms are consistent with a minimization problem. Thus, minimizing the objective function solves the optimization problem.
  • the credit would be the average air temperature within the room volume.
  • the constraint would be a limit on the point-wise temperature distribution within the room, which, in the form of a penalty term, would be calculated as the average temperature violation.
  • the penalty term is the maximum value of the point-wise temperature violations within the room.
  • the form of the generic objective function thus allows any number of credit and penalty terms to be defined in a general manner for the problem being solved.
  • Forms for the credit or penalty terms include, but are not limited to:
  • Fig. 1 illustrates an embodiment of an architecture according to the present invention.
  • a server 10 includes a graphical user interface 12 connected to a processor 14.
  • the processor 14 is connected to a memory 16.
  • the server 10 is directly accessible by a user input device 18 (e.g., a display, keyboard and mouse).
  • the server 10 is also accessible by computers 22 and 26 over an intranet 20 and the Internet 26, respectively.
  • a user input device 18 e.g., a display, keyboard and mouse
  • computers 22 and 26 over an intranet 20 and the Internet 26, respectively.
  • the operation of the architecture shown in Fig. 1 will be discussed in detail below.
  • a configured objective function satisfying the above-described generic definition is already stored in the memory 16 of the server 10.
  • the configured objective function could have been configured according to one of the embodiments described below.
  • the user instructs the server 10 to provide a list of the configured objective functions stored in the memory 16, and instructs the server 10 to use one of the listed configured objective functions.
  • a user via input 18, computer 26 or computer 22 accesses the server 10 over the graphical user interface 12.
  • the user supplies the server 10 with a configured objective function meeting the definition of the above-described generic definition.
  • the user supplies the configured objective function using any well-known programming language or program for expressing mathematical expressions.
  • the user instructs the processor 14 via the graphical user interface 12 to upload a file containing the configured objective function.
  • the processor 14 then uploads the file, and stores the file in memory 16.
  • configuring the objective function is interactive between the user and the server 10.
  • the user instructs the processor 14 to start the process for configuring an objective function.
  • the processor 14 requests the user to identify the number of credit components and the number of penalty components.
  • the processor 14 requests that the user provide a mathematical expression for the credit term and an initial weight for the associated credit weight.
  • the processor 14 requests that the user provide a mathematical expression for the penalty term and an initial weight for the associated penalty weight.
  • the processor 14 via the graphical user interface 12 accepts definitions of mathematical expressions according to any well-known programming language or program.
  • the server 10 is preprogrammed for use on a particular constraint or optimization based problem.
  • the server 10 stores possible optimization parameters and possible constraint parameters associated with the particular optimization or constraint problem.
  • the processor 14 accesses the possible optimization parameters already stored in the memory 16, and provides the user with the option of selecting one or more of the optimization parameters for optimization.
  • Fig. 2 illustrates a screen shot of a optimization configuration page used in selecting one or more optimization parameters associated with the optimization problem of boiler water reactor core design according to this embodiment of the present invention.
  • the optimization parameters 40 of optimize rod patterns, optimize core flow, and optimize sequence intervals are available for selection by the user as optimization parameters.
  • Optimize rod patterns means making an optimal determination of individual control rod positions within a control rod grouping (called a sequence), for the duration of time during the operating cycle when the given sequence is being used to control the reactor.
  • Rod positions affect the local power as well as the nuclear reaction rate.
  • Optimize core flow means making an optimal determination of reactor coolant flow rate through the reactor as a function of time during the operating cycle. Flow rate affects global reactor power as well as the nuclear reaction rate.
  • Optimize sequence intervals means making an optimal determination of the time duration a given sequence (i.e., control rod grouping) is used to control the reactor during the operating cycle. Sequence intervals affect local power as well as the nuclear reaction rate.
  • the user selects one or more of the optimization parameters by clicking in the selection box 42 associated with an optimization parameter 40.
  • a check appears in the selection box 42 of the selected optimization parameter. Clicking in the selection box 42 again de-selects the optimization parameter.
  • the memory 16 also stores constraint parameters associated with the optimization problem.
  • the constraint parameters are parameters of the optimization problem that must or should satisfy a constraint or constraints.
  • Fig. 3 illustrates a screen shot of a optimization constraints page listing optimization constraints associated with the optimization problem of boiler water reactor core design according to this embodiment of the present invention.
  • each optimization constraint 50 has a design value 52 associated therewith.
  • Each optimization constraint must fall below the specified design value.
  • the user has the ability to select optimization parameters for consideration in configuring the objective function.
  • the user selects an optimization constraint by clicking in the selection box 54 associated with an optimization constraint 50. When selected, a check appears in the selection box 54 of the selected optimization constraint 50. Clicking in the selection box 54 again de-selects the optimization constraint.
  • Each optimization parameter has a predetermined credit term and credit weight associated therewith stored in memory 16.
  • each optimization constraint has a predetermined penalty term and penalty weight associated therewith stored in memory 16.
  • the penalty term incorporates the design value, and the user can change (i.e., configure) this value as desired.
  • the embodiment of Fig. 3 allows the user to set an importance 56 for each optimization constraint 50.
  • the importance field 58 for an optimization constraint the user has pull down options of minute, low, nominal, high and extreme. Each option correlates to a predetermined penalty weight such that the greater the importance, the greater the predetermined penalty weight. In this manner, the user selects from among a set of predetermined penalty weights.
  • the processor 14 configures the objective function according to the generic definition discussed above and the selections made during the selection process.
  • the resulting configured objective function equals the sum of credit components associated with the selected optimization parameters plus the sum of penalty components associated with the selected optimization constraints.
  • Fig. 4 illustrates a flow chart of an optimization process employing the adaptive determination of weighting factors according to the present invention.
  • the optimization process of Fig. 4 will be described as being implemented by the architecture illustrated in Fig. 1. Accordingly, this process is performed when a user instructs the server 10 to perform such a process via input device 18, computer 22 or computer 26.
  • the objective function is configured as discussed above in the preceding section, then the optimization process begins.
  • the processor 14 retrieves from memory 16 or generates one or more sets of values for input parameters (i.e., system inputs) of the optimization problem based on the optimization algorithm in use.
  • Some of the input parameters would be placement of fresh and exposed fuel bundles within the reactor, selection of the rod groups (sequences) and placement of the control rod positions within the groups as a function of time during the cycle, core flow as a function of time during a cycle, reactor coolant inlet pressure, etc.
  • Each input parameter set of values is a candidate solution of the optimization problem.
  • the processor 14 runs a simulated operation and generates a simulation result for each input parameter set of values. For example, for boiler water reactor core design, a well-known simulation program for boiler water reactor operation is run using an input parameter set.
  • the simulation result includes values (i.e., system outputs) for the optimization parameters and optimization constraints. These values, or a subset of these values, are values of the variables in the mathematical expressions of the objective function.
  • step S14 and S16 the processor 14 uses an objective function and the system outputs to generate an objective function value for each candidate solution.
  • the objective function of step S16 includes the initial credit and penalty weights established when the objective function was configured. These initial credit and penalty weights are referred to as the static weights.
  • the objective function of step S14 includes adapted penalty and credit weights, which were adapted in a previous iteration as discussed in detail below with respect to steps S22 and S24.
  • step S18 the processor 14 assess whether the optimization process has converged upon a solution using the objective function values generated in step S14. If convergence is reached, then the optimization process ends.
  • step S22 the processor 14 determines whether to perform adaptation of the credit and penalty weights based on the current iteration. For example, in one embodiment, the weight adaptation is performed at a predetermined interval (e.g., every five iterations). In another embodiment, the weight adaptation is performed at predetermined iterations. In a still further embodiment, the weight adaptation is performed randomly.
  • step S24 weight adaptation is performed as shown in Fig. 5.
  • the processor 14 determines the value of the penalty components in the objective function using the static weights (i.e, the weights initially assigned to the credit and penalty components) and system outputs for the candidate solution producing the best objective function value when the adaptive weights are applied (Many optimization search algorithms proceed based on the candidate solution producing the best objective function value. Other optimization algorithms pursue more than one objective function value at a time.)
  • step S52 the processor 14 determines whether any constraint violations exist based on the determinations made in step S50.
  • a constraint violation exists when any penalty component is positive. If a constraint violation exists, then in step S54, the processor 14 determines the worst penalty component (greatest positive value) based on the determinations made in step S50. Then in step S56, the penalty weight for the worst penalty component is increased. In one embodiment, the increment is achieved by multiplying the penalty weight by a predetermined constant ⁇ that is greater than or equal to one. Also, in step S56, the weights of the other penalty components and the credit components are decreased. In one embodiment, the decrement is achieved by multiplying the weights by a predetermined constant ⁇ that is less than or equal to one. After step S56, processing proceeds to step S26 of Fig. 4.
  • incrementing and decrementing techniques in step S56 are not limited to multiplication.
  • incrementing and decrementing can be performed using any well-known mathematical operation.
  • decrementing is not limited to decrementing the credit and penalty weights by the same amount or factor, and the incrementing and decrementing can change based on the iteration.
  • step S58 the processor increments the credit weight for each credit component. In one embodiment, the increment is achieved by multiplying the penalty weight by a predetermined constant ⁇ that is greater than or equal to one. After step S58, processing proceeds to step S26 of Fig. 4.
  • step S58 is not limited to multiplication.
  • incrementing can be performed using any well-known mathematical operation.
  • the incrementing is not limited to incrementing each of the credit weights or incrementing the credit weights by the same amount or factor. And the incrementing can change based on the iteration.
  • step S26 the input parameter sets are modified, the optimization iteration count is increased and processing returns to step S12.
  • the generation, convergence assessment and modification operations of steps S12, S18 and S26 are performed according to any well-known optimization algorithm such as Genetic Algorithms, Simulated Annealing, and Tabu Search.
  • the optimization algorithm can be, for example, one of the optimization processes as disclosed in U.S. Application No. 09/475,309, titled SYSTEM AND METHOD FOR OPTIMIZATION OF MULTIPLE OPERATIONAL CONTROL VARIABLES FOR A NUCLEAR REACTOR filed December 30, 1999 or U.S. Application No. 09/683,004, tilted SYSTEM AND METHOD FOR CONTINUOUS OPTIMIZATION OF CONTROL-VARIABLES DURING OPERATION OF A NUCLEAR REACTOR, filed November 7, 2001.
  • the invention provides a systematic and general method for handling optimality conditions and constraints within the context of the penalty function approach for Constrained Optimization Problems (COPs) and Constraint Satisfaction Problems (CSPs), independent of the optimization search employed. Because of the flexibility of the invention, changes in optimality conditions, constraint term definitions, and adaptive parameter definitions are readily accommodated. Because the worst penalty component is calculated from the product of the penalty weight and the penalty term, where the penalty weights are the initial static values, for example, as input by the user; this adaptive methodology provides memory tying the objective function back to the original starting point. Also, in an alternative embodiment, the values of the adaptive and static objective function, as they change over time, are displayed as a plot or graph for the user on the input device 18, the computer 22 or the computer 26. In this manner, the user can see the static measure of the optimization progress.
  • COPs Constrained Optimization Problems
  • CSPs Constraint Satisfaction Problems
  • the technical effect of the invention is a computer system that provides for adapting the objective function, used in determining the progress of an optimization operation, by adaptively adjusting the weight factors of the components forming the objective function.

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EP03255772A 2002-09-19 2003-09-16 Procédé et dispositif pour la détermination adaptive de facteurs de pondération dans le contexte d'une function objective Ceased EP1400905A1 (fr)

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US20040059696A1 (en) 2004-03-25
TW200413869A (en) 2004-08-01
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US7487133B2 (en) 2009-02-03

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